The strong Pytkeev property for topological groups and topological vector spaces

Generalizing the notion of metrisability, recently Tsaban and Zdomskyy (Houst J Math 35:563–571, 2009) introduced the strong Pytkeev property and proved a result stating that the space (formula presented) has this property for any Polish space (formula presented). We show that the strong Pytkeev property for general topological groups is closely related to the notion of a (formula presented)-base, investigated in Gabriyelyan et al. (On topological groups with a small base and metrizability, preprint) and Ka̧kol et al. (Descriptive Topology in Selected Topics of Functional Analysis. Developments in Mathematics. Springer, Berlin, 2011). Our technique leads to an essential extension of Tsaban–Zdomskyy’s result. In particular, we prove that for a Čech-complete (formula presented) has the strong Pytkeev property if and only if (formula presented) is Lindelöf. We study the strong Pytkeev property for several well known classes of locally convex spaces including (formula presented)>-spaces and strict (formula presented)-spaces. Strengthening results from Cascales et al. (Proc Am Math Soc 131:3623–3631, 2003) and Dudley (Proc Am Math Soc 27:531–534, 1971) we deduce that the space of distributions (formula presented) (which is not a (formula presented)-space) has the strong Pytkeev property. We also show that any topological group with a (formula presented)-base which is a (formula presented)-space has already the strong Pytkeev property. We prove that, if (formula presented)-space, then the free abelian topological group (formula presented) and the free locally convex space (formula presented) have the strong Pytkeev property. We include various (counter) examples and pose a dozen open questions.